"TIFF" is the "Tagged Image File Format"
TIFF is a very flexible image format that allows for images and attribute (or data) to be stored in a single file. The images can be stored efficiently, but with some loss in quality, or they can be stored perfectly though it creates a much larger file.
TIFF is a bit unusual in that it can store more than one image in a file while most image formats store only one.
TIFF can also store data with much greater precision, if the program creating the images is using such precision (say 10bits / channel instead of the typical 8).
TIFF can also encode vector data which manipulate your image. An example of this would be if you cropped your image using an elliptical path. TIFF would save the underlying image, the clipping path, and a TIFF-compatible reader would be able to deliver your clipped image later.
Finally, the "tagged" part of TIFF allows other data to come along with the file. If you have a bunch of Photoshop settings or if you've annotated your image with your name, address and company, then TIFF can "remember" that information along with your images.
Unless you have unusual needs, TIFF is generally a wasteful way of transferring data. PNG and Jpeg are generally your best bets for image transmission.
If you want to try TIFF anyway, just "Save As..." and select the TIFF file format.